A318497 Numerators of the sequence whose Dirichlet convolution with itself yields A061389, number of (1+phi)-divisors of n.

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 13, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, -13, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 13, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 1

Author

Antti Karttunen, Aug 30 2018

Keywords

• sign
• frac

Comments

No zeros among the first 2^20 terms. This is most probably multiplicative, like A318498.

Links

• Antti Karttunen, Table of n, a(n) for n = 1..65537

Crossrefs

Cf. A061389, A318314 (denominators).

Programs

PARI

up_to = 65537;
A061389(n) = factorback(apply(e -> (1+eulerphi(e)),factor(n)[,2]));
DirSqrt(v) = {my(n=#v, u=vector(n)); u[1]=1; for(n=2, n, u[n]=(v[n]/v[1] - sumdiv(n, d, if(d>1&&dA317937.
v318497_98 = DirSqrt(vector(up_to, n, A061389(n)));
A318497(n) = numerator(v318497_98[n]);

Formula

a(n) = numerator of f(n), where f(1) = 1, f(n) = (1/2) * (A061389(n) - Sum_{d|n, d>1, d 1.